Algebra and Combinatorics Seminar

Date: April 9, 2021

Time: 3:00PM - 4:00PM

Location: Zoom

Speaker: Anton Dochtermann, Texas State University

Title: Betti numbers of random edge ideals

Abstract: We study asymptotic homological properties of random quadratic monomial ideals in a polynomial ring R = k[x_1, . . . , x_n], utilizing methods from the Erd\"os-R\'enyi model of random graphs. Here we consider a graph on n vertices and exclude an edge (corresponding to a quadratic generator of the ideal I) with probability p, and consider algebraic properties as n tends to infinity. Our main results involve fixing the edge parameter p = p(n) so that asymptotically almost surely the Krull dimension of R/I is fixed. Under these conditions we establish various properties regarding the Betti table of R/I, including sharp bounds on regularity and projective dimension and distribution of nonzero Betti numbers. These results extend work of Erman-Yang, who studied such ideals in the context of conjectured phenomena in the nonvanishing of asymptotic syzygies. Our results use collapsibility properties of random clique complexes and Garland's method regarding spectral gaps of graphs, and in particular rely on the underlying field in some cases. This is joint work with Andrew Newman.